When the polynomial r(x) = ax + b is added to the polynomial p(x) = 4x⁴ + 2x³ - 2x² + x − 1, then the resulting polynomial is divisible by the polynomial q (x) = x² + 2x - 3. What is the value of (a - b)?

Given:

r(x) = ax + b

p(x) = 4x⁴ + 2x³ - 2x² + x - 1

q(x) = x² + 2x - 3

p(x) + r(x) is divisible by q(x)

Formula used:

If a polynomial f(x) is divisible by g(x), then the roots of g(x) are also roots of f(x).

Calculation:

The roots of q(x):

q(x) = x² + 2x - 3 = (x + 3)(x - 1)

Roots of q(x) are x = -3 and x = 1.

Now,

p(x) + r(x) = 4x⁴ + 2x³ - 2x² + x - 1 + ax + b

⇒ 4x⁴ + 2x³ - 2x² + (a + 1)x + (b - 1)

Since p(x) + r(x) is divisible by q(x), the roots of q(x) are also roots of p(x) + r(x).

Substitute x = 1:

4(1)⁴ + 2(1)³ - 2(1)² + (a + 1)(1) + (b - 1) = 0

4 + 2 - 2 + a + 1 + b - 1 = 0

4 + a + b = 0

a + b = -4 ...(1)

Substitute x = -3:

4(-3)⁴ + 2(-3)³ - 2(-3)² + (a + 1)(-3) + (b - 1) = 0

4(81) + 2(-27) - 2(9) - 3a - 3 + b - 1 = 0

324 - 54 - 18 - 3a - 4 + b = 0

248 - 3a + b = 0

-3a + b = -248 ...(2)

Solve equations (1) and (2) for a and b:

Subtract (1) from (2):

(-3a + b) - (a + b) = -248 - (-4)

-4a = -244

a = 61

Substitute a = 61 into (1):

61 + b = -4

b = -65

Now,

a - b = 61 - (-65) = 61 + 65 = 126

∴ The value of (a - b) is 126.